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Maxence BIGERELLE, Alain IOST - Statistical artefacts in the determination of the fractal dimension by the slit island method - Engineering Fracture Mechanics - Vol. 71, n°7-8, p.1081-1105 - 2004
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Science Arts & Métiers (SAM)
is an open access repository that collects the work of Arts et Métiers ParisTech
researchers and makes it freely available over the web where possible.
This is an author-deposited version published in: http://sam.ensam.eu
Handle ID: .http://hdl.handle.net/null
To cite this version :
Maxence BIGERELLE, Alain IOST - Statistical artefacts in the determination of the fractal dimension by the slit island method - Engineering Fracture Mechanics - Vol. 71, n°7-8, p.1081-1105 - 2004
Any correspondence concerning this service should be sent to the repository Administrator : archiveouverte@ensam.eu
Statistical artefacts in the determination of the
fractal dimension by the slit island method
Maxence Bigerelle, Alain Iost
*Equipe ENSAM ‘‘Surfaces et Interfaces’’, Laboratoire de Meetallurgie Physique et Geenie des
Mateeriaux, CNRS UMR 8517, 8 Boulevard Louis XIV, 59046 Lille Cedex, France
Abstract
This paper comments upon some statistical aspects of the slit island method which is widely used to calculate the fractal dimension of fractured surfaces or of materialsÕ features like grain geometry. If a noise is introduced when measuring areas and perimeters of the islands (experimental errors), it is shown that errors are made in the calculation of the fractal dimension and more than a false analytical relation between a physical process parameter and the fractal dimension can be found. Moreover, positive or negative correlation with the same physical process parameter can be obtained whether the regression is performed by plotting the variation of the noisy area versus the noisy perimeter of the considered islands or vice versa. Monte-Carlo simulations confirm the analytical relations obtained under statistical considerations.
Keywords: Fractal dimension; Slit island method; Monte-Carlo simulation; Statistics; Artefact measurement
1. Introduction
Since MandelbrotÕs [1,2] pioneering work from the fractal geometry has been extensively applied to describe various irregular phenomena in nature. As it was shown that real surfaces such as fractured surfaces in materials are fractal [3–5] i.e. self-similar (or self-affine) over a wide range of scales, extensive work was done to correlate the fractal dimension of the surfaces with mechanical properties such as impact energy [6], fracture toughness [7], fatigue crack propagation [8]. . . The experimental results show that a general conclusion cannot easily be drawn [9]: some studies report a positive variation of fracture toughness along with the fractal dimension [6,10–13], and others a negative one [14–17]. To other researchers, there is either no correlation [18–20] or the fractal dimension of the fractured surfaces is a universal constant [21]. One of the major reasons to this discrepancy is related to the method used to calculate the fractal di-mension. Different methods (Slit Island [2,14], Vertical Section [22], Richardson [23], Minkowski Sausage
*
Corresponding author. Tel.: +33-3-2062-2233; fax: +33-3-2062-2957.
[24], Box Counting [25,26], Spectrum [26,27]. . .) applied onto the same fractured surface can give different fractal dimensions. If some discrepancies can easily be understood by the experimental limits and material properties (length of the recorded profile, textured material, structural heterogeneity. . .), others seem re-lated to the method used to calculate the fractal dimension.
The aim of this paper is to analyse the slit island method (SIM), one of the most widely used to determine the fractal dimension of fractured surfaces, D, and to show that, in some cases, the origin of positive or negative correlation between fractal dimension and mechanical properties may be caused by statistical artefacts related to noise measurement.
2. The slit island method
For Euclidean forms, the ratio between perimeter (P ) and the square root of the area (A) is an a-dimensional constant (e.g. 2pffiffiffip for a circle). This ratio, known as the shape factor, is extensively used in quantitative metallography. However for fractal shapes such as cloud [28] mammalÕs cerebral cortex [29], human lung [30], basins of attraction [31], this ratio is not constant but depends on the observation scale. From these observations, Mandelbrot [2] derived a perimeter–area relationship which when applied to fractured surfaces is called the ‘‘Slit Island Method’’. Mandelbrot, Passoja and Paullay first applied the method in 1984 [14] when studying the fracture toughness of 300-grade maraging steel and correlated the fractal dimension with the impact energy. It consists in electroplating and mounting in resin the fractured specimen in order to ensure edge retention. The specimen is then polished parallel to the plane of fracture in successive stages. At varying intervals of grinding time corresponding to a few micrometers removal, the sample is observed in optic or scanning electron microscopy. Islands of the material first merged in the resin and grow in size as polishing progresses. The islands contain ‘‘lake within islands and islands within lakes’’. Mandelbrot et al. [14] include the former and neglect the latter, but different solutions are adopted by other authors [32,33].
This method was founded on the following statements:
ii(i) When islands are derived from initial self-affine fractal surface of dimension Dsby sectioning with a plane, their coastlines are self-similar fractals with dimension D¼ Ds 1.
i(ii) The relation between perimeter and area is given by Eq. (1): RðgÞ ¼½P ðgÞ
1=D ffiffiffiffiffiffiffiffiffiffi AðgÞ
p ð1Þ
where RðgÞ is a constant which depends on the choice of the yardstick length, g, used to measure the length along the walking path.
This equation is only true for self-similar islands [2,14,34,35] whose perimeter and area are measured in the same way.
(iii) When the graph of log(Perimeter) versus log(Area) is rectilinear, the fractal dimension is deduced from the slope.
In spite of a few serious problems with the method [3,19,32,34–39] a lot of papers have been published over the past 10 years concerning the relationship between mechanical properties and the fractal dimension calculated by the SIM. The most surprising result was an inversion relation between fractal dimension and toughness for the same material [6,14] that allows us to investigate if all the criticisms against the method were not due to some statistical errors. For this reason, in the following paragraph, we shall consider only 1082
the influence of an experimental noise on the validity of the calculation of the fractal dimension and the related correlation with mechanical properties.
3. Statistical aspect of the area–perimeter relation
The perimeter area relation Eq. (1) could be written in the following form: logðAÞ ¼ 2
DlogðP Þ þ b ð2Þ
and the fractal dimension is obtained from the slope of the plot of log(Area) versus log(Perimeter). The precision depends on the density probability of both the area and the perimeter. Different shapes obtained by the SIM on samples from the same material get an intrinsic variation. The distribution was defined by Mandelbrot as a stochastic self-similarity [2], which means that both the perimeter and the area measurements get a probability density function (PDF) (even if this PDF is unknown) and as a consequence an intrinsic variation.
3.1. Determination of the uncertainty
To apply SIM, all islands must be measured with the same yardstick length g [2,40].
Let n, be the number of islands, respectively Pithe perimeter and Aithe area of the ith island, a¼ 2=D, the slope of the linear plot of logðAÞ ¼ f ½logðP Þ and a its estimation.
According to Eq. (1), the fractal dimension is estimated by linear regression of the relation: logðAÞ ¼ 2
DlogðP Þ þ b þ e ð3Þ
where b is the intercept at the origin, and e a random vector that represents the noise on the area mea-surement.
If sA is the unbiased estimation of the residual standard deviation then: s2A¼ Pn i¼1ðlog A Model i log A Exp: i Þ 2 n 2 ð4Þ
Given the following hypotheses:
Hypothesis 1. e obeys a Gaussian law meaning that the measure of the area obeys a lognormal law. Hypothesis 2. The residual variance is constant independently of the value of logðP Þ (homoscedasticity). Hypothesis 3. The e autocorrelation function equals zero.
Then the variable t¼ ða aÞ=sa obeys here a Student law with n 2 degrees of freedom and the a variance (variance of the slope given by Eq. (3)) is given by:
s2 a¼
s2 A Pn
i¼1ðlog Pi log P Þ
2 ð5Þ
where P is the mean of the island perimeter:Pni¼1ðlog Pi log P Þ 2
¼ ðn 1ÞVarðlog P Þ. If the additional hypotheses are considered:
Then if Pmin and Pmax are respectively the minimum and the maximum of the perimeter range of all recorded islands, then:
Varðlog P Þ ¼ðlog Pmax log PminÞ 2 12 and therefore ra¼ ffiffiffiffi s2 a q ¼ se ffiffiffiffiffi 12 p ðlog Pmax log PminÞ
ffiffiffiffiffiffiffiffiffiffiffi n 1 p
where seis the standard deviation of the noise e given by Eq. (3).
Taken db the required accuracy on the determination of the slope (at b¼ 95% confidence interval): 2tn2se ffiffiffiffiffi 12 p DP ffiffiffiffiffiffiffiffiffiffiffi n 1 p 6db ð6Þ
with DP ¼ logPPmaxmin, the number of decades used for the determination of the perimeter. Hypothesis 5. The number of islands is higher than 30.
Then the Student law converges towards a Laplace–Gauss one and tn>30 2, then: 4se ffiffiffiffiffi 12 p DP ffiffiffin p 6db ð7Þ
Hypothesis 4b. The area A obeys a uniform PDF.
Instead of using Eq. (3), the regression to calculate D is carried out with Eq. (8): logðP Þ ¼D
2 logðAÞ þ b
0þ e0 ð8Þ
where e0represents the noise on the perimeter measurement and b0 the intercept. With the same reasoning as above the variance for the fractal dimension is:
4se0 ffiffiffiffiffi12 p DA ffiffiffin
p 6db0 ð9Þ
where DA is the number of decades used for the determination of the area (DA¼ logAAmax min). The following statistical considerations can be stated:
ii(i) The island-recorded area must cover a broad range of sizes since the islands obtained from experimen-tal images are random variables whose scattering depends on the recording process. If the experimenexperimen-tal range used to calculate the fractal dimension is not large enough, so the noise due to the error during measurement is too large and no correlation can be found. The SIM does not apply, but perhaps the Richardson method [41] can give more reliable results.
i(ii) The precision in the determination of the fractal dimension is better if the number of islands considered is high.
(iii) To double the precision in the determination of the fractal dimension we must increase fourfold the number of islands or double the number of decades.
3.2. Area versus perimeter or perimeter versus area?
Positive or negative fractal dimension values were found by Ray et Mandal [6] using respectively pe-rimeter or area in the abscissa for the log–log plot. In the same way, the analysis of the results coming from the literature shows that the positive or negative correlation between D and the mechanical properties (or the lack of correlation) depends on the abscissa (log P or log A) chosen for the regression (Table 1) [42]. These results obviously question the validity of the Slit Island Method. To answer these questions we first studied the precision in the determination of the fractal dimension related to the two types of represen-tation.
3.2.1. Perimeter versus area representation
With Hypotheses 1, 2, 3 and 5, when D is deduced from Eq. (8), the density function of the residual obeys a Gaussian law as well as a (from Eq. (3)) but not D since the change of variable a¼ 2=D destroys the normality of the estimation of the fractal dimension.
If DP¼f ðAÞand DA¼f ðP Þare the fractal dimensions obtained respectively by Eqs. (8) and (3), we will prove that EðDP¼f ðAÞÞ is different from EðDA¼f ðP ÞÞ where EðX Þ is the expectation of the variable X .
Demonstration: If a obeys a Gaussian law then
x¼ 2 DA¼f ðP Þ aa
ra
is a reduced centred Gaussian and then: DA¼f ðP Þ¼ 2 a a 1 1þra a ax
Developing in function of the x rising order:
DA¼f ðP Þ¼ 2 a a 1 " ra a a xþ ra a a 2 x2 ra a a 3 x3þ # Table 1
Literature results of impact or fracture toughness measurements related to the fractal dimension
Author Reference Relation DP DA sP sA nisl Dmin Dmax DD nD Corr.
Mandelbrot [14] A¼ f ðP Þ 3 4 0.15 0.2 48 1.1 1.3 0.2 6 –
Su [15] P¼ f ðAÞ 0.6 0.6 0.1 0.05 31 1.2 1.8 0.6 5 –
Pande [18] P¼ f ðAÞ 1 2 0.1 0.15 20 1.41 1.46 0.06 5 ¼¼¼
Wang (Fat) [8] P¼ f ðAÞ 2.5 4 0.07 0.17 44 1.1 1.22 0.12 9 +++
Ray [6] A¼ f ðP Þ 2 3 0.2 0.2 15 1 1.52 0.52 5 +++
Su [44] P¼ f ðAÞ 0.7 1.1 0.05 0.04 41 1.15 1.37 0.22 4 +++
Mecholski [11] A¼ f ðP Þ 1.4 2.2 0.22 0.21 27 1.04 1.2 0.16 11 +++
Pande [45] A¼ f ðP Þ 1.4 2.2 0.1 0.16 37 1.32 1.32 1 ¼¼¼
Richards [19] A¼ f ðP Þ 2 3 0.07 0.27 23 1.77 1.91 0.14 4 ¼¼¼
DP: Number of decades of perimeter, DA: Number of decades of area, sP: Standard deviation of the perimeter measure (in log), sA:
Standard deviation of the area measure (in log), nisl: Number of islands, Dmin: Lower fractal dimension, Dmax: Higher fractal dimension,
DD: Range variation of D, nD: Number of samples used to quantify the relation.
+++: Means positive correlation,¼¼¼: No correlation, – : Negative correlation. (Italic: fatigue crack profiles and relation between
and calculating the mean E½DA¼f ðP Þ from DA¼f ðP Þ: E½DA¼f ðP Þ ¼ 2 a a 1 " þ ra a a 2 l2P¼f ðAÞþ ra a a 4 l4P¼f ðAÞþ # ð10Þ where lN
P¼f ðAÞ is the moment of order N of the fractal dimension calculated by Eq. (8). The expected value exists only if the preceding series converges and then
lim i!1 l2iþ2 P¼f ðAÞ l2i P¼f ðAÞ ! <1 However, from Eq. (10),
lim i!1 l2iþ2P¼f ðAÞ l2i P¼f ðAÞ ! >1
and then the series diverges and the fractal dimension cannot be calculated by means of Eq. (3) (see Ap-pendix A). This result is understandable since the probability is not nul to obtain a zero slope by using Eq. (3) and then an infinite fractal dimension. However, it is possible to state positively that if the experimental noise is low then a > 0, meaning that the Gaussian law for a is truncated. With the assumption that a2 ½aa T ; aaþ T with aa T 0, lim i!1 l2iþ2 P¼f ðAÞ l2i P¼f ðAÞ ! ¼ T2
and E½DA¼f ðP Þ converges ifðra a aÞ 2 T2<1. More if r a=aa is low: E½DA¼f ðP Þ 2 a a 1 " þ ra a a 2 l2P¼f ðAÞ # ð11Þ and it appears a positive biasðra
a aÞ 2 l2 P¼f ðAÞ.
If T is high enough (the a Gaussian is weakly truncated) then l2
P¼f ðAÞ 1. If aais perfectly determined then aa¼ 2=D, and the following fundamental equation can be obtained:
E½DA¼f ðP Þ D 1 " þ raD 2 2# ð12Þ This equation involves the following physical consequences:
ii(i) If the residual scatter for the linear plot of log(Area) versus log(Perimeter) is low, then E½DA¼f ðP Þ 2=aa and therefore there is no difference for the fractal dimension calculated by means of Eq. (8) or Eq. (3). i(ii) The fractal dimension is especially overestimated when the true fractal dimension is high.
(iii) The fractal dimension is especially overestimated when the experimental noise rises.
Statistical biases are often reported in material experimental measurements that lead to misinterpre-tations when neglected [43].
(iv) Suppose that the real unknown fractal dimension D is unchanged, then from Eq. (12): • As r2
a/ 1=n, when the number of recorded islands rises, the estimated fractal dimension diminishes even if the true value is constant.
• As r2 a/ s
2
e, the calculated fractal dimension rises with the residual variance. • As r2
a/ 1=DP, the fractal dimension rises in proportion with the number of decades. 1086
From these relations it is clear that if a physical process does not change the fractal dimension, but modifies the image morphology with a monotonous law (fewer islands, different noise levels) a correlation between the fractal dimension and the physical process can be erroneously stated.
This statistical artefact may be at the origin of some correlations between mechanical properties (MP) and the fractal dimension reported in literature. If the fractal dimension of the fractured surface does not depend on MP, Eq. (3) implies a negative correlation between fractal dimension and MP if:
ii(i) The number of recorded islands rises as MP rises. i(ii) The area standard deviation decreases with MP. (iii) The number of decades rises with MP.
On the other hand provided the Hypotheses 1–3 are respected, the average calculated fractal dimension tends toward the theoretical one when Eq. (8) is used.
3.2.2. Area versus perimeter representation
Different fractal dimensions are obtained when using the regression of log(Area) versus log(Perimeter) or log(Perimeter) versus log(Area). In fact, the least square theory leads to the following equations without any prior hypotheses related to the experimental data:
DP¼f ðAÞ¼ 2r rP rA ð13Þ DA¼f ðP Þ¼ 2 1 r rP rA ð14Þ where rP and rA are respectively the standard deviation for log(Perimeter) and log(Area) and r the cor-relation coefficient. From the above equations it can be stated that:
DP¼f ðAÞ DA¼f ðP Þ
¼ r2 ð15Þ
As a consequence the same fractal dimension is obtained for the two types of representation only if the mathematical relationship is perfect i.e. the coefficient of correlation is r¼ 1. As this coefficient does not depend on the regression, the difference between the fractal dimension calculated by means of Eqs. (3) and (8) rises with the scattering of the data. It is usual to take as abscissa the variable X which is controlled by the experimenter, and as ordinate the answer Y (dependent variable that contains experimental noise). For the Richardson plots the unambiguous abscissa is the yardstick length that is a deterministic value and Y the measured perimeter that is a stochastic value. For the slit island method there is no a priori reason for choosing perimeter or area in ordinate.
Nevertheless Eq. (15) requires the knowledge of the regression coefficient, which depends on the ex-perimental data, and it is necessary to evaluate the discrepancy.
3.2.3. Probabilistic behaviour
Assuming that the measure of the area is obtained precisely, DP¼f ðAÞis calculated such as logðP Þ ¼ c logðAÞ þ b0þ e0; with c¼DP¼f ðAÞ
2 As
r2¼ 1 s 2 e0
Var logP with
DP¼f ðAÞ DA¼f ðP Þ
where se0is the standard deviation of residual e0(such that Eðse0Þ ¼ 0 where EðxÞ is the mean of x). Var logðP Þ can be split up with a modelled and a residual variance Var logP ¼ Var logP jmodþ s2
e0. log Pjmod¼ c log A þ b induces Var logPjmod¼ c2Var logA (supposing covðc; bÞ 0) and if log A obeys a uniform law with interval DA, then Var logPjmod¼ c
2 D2A 12, and as a consequence: DP¼f ðAÞ DA¼f ðP Þ ffi 1 s 2 e0 s2 e0þ c2 D 2 A 12 ð16Þ
This formula entails the following remarks:
i(i) DA¼f ðP Þ>DP¼f ðAÞ the fractal dimension calculated by Eq. (3) is always higher than that calculated by Eq. (8).
(ii) The difference between these two methods decreases when: • The fractal dimension rises because c ¼ D=2.
• The number of decade rises.
• The noise in the perimeter measurement decreases.
3.2.4. Statistical behaviour
The above considerations are only probabilistic, and sampling is not considered. Because of the sta-tistical bias, results can be different from the solutions of Eq. (16). To test whether the experimental data match the model corresponding to Eq. (16), an attempt to theoretical approach has been performed, but no simplification for the density function was obtained (the moment is not defined for DA¼f ðP Þ, then the r2 density function cannot be simplified). For this reason, we proceeded a Monte-Carlo simulation as follows on purpose to analyse Eq. (16):
Let @a;s;DA;k;n;t; be the space of configurations, c2 f0:5; 0:6; . . . ; 2g; se02 f0:01; 0:02; . . . ; 0:5g: DA2 f0:5; 0:1; . . . ; 5g; and A 2 f0 nDA; 1 nDA; 2 nDA; . . . ; k nDA; . . . ; n
nDAg where DA is the decade number in the area measurement, n is the number of data used for the regression analysis and se0 the standard deviation of the noise in the area measurement (in log). This set of configurations is based on experimental values found in literature for the SIM.
The relation logðP Þ ¼ a logðAÞ þ e0 is simulated 100 times, where error e0 simulated by the Cox and Muller transformation obeys a Gaussian law with zero mean and standard deviation se0.
The following notations are adopted: Let Pa¼ha;s¼hsa;DA¼hDA;k¼hk;n¼hn;t¼ht be an element of @a;s;DA;k;n;t; and (Aa¼ha;s¼hsa;DA¼hDA;k¼hk;n¼hn;t¼ht; Pa¼ha;s¼hsa;DA¼hDA;k¼hk;n¼hn;t¼ht) the co-ordinates of a simulated point with t the tth simulation.
The index h will be substituted by h¼ when the pair is considered for all discrete values h of @. In the same way fðPa¼ha;s¼hs;DA¼hDA;k¼hk;n¼hn;t¼htÞ is a statistic on the space a ¼ ha; s¼ hs;DA¼ hDA; k¼ hk; n¼ hn; t¼ ht, and fðPa;s;DA;k;n;tÞ a statistic built on @ and with value on @.
Examples:
• Pa¼ha;s¼hs;DA¼hDA;k¼hk;n¼100;t¼ is the space of 100 pairs simulated with a¼ ha; s¼ hs;DA¼ hDA; k ¼ hk, and the associated mean is E½Pa¼ha;s¼hs;DA¼hDA;k¼hk;n¼100;t¼.
• E½Pa;s;DA;k;n¼100;t¼ is the mean of the area and perimeter calculated for 100 Monte-Carlo simulations for each value of the parameters a, s, DA, k members of@.
We first verified Eq. (16) by simulating Pa;s;DA;k;n¼100;tand the calculation of the regression slopes obtained by the least square method for Eqs. (3) and (8). The mean
E D P¼f ðAÞ a;s;DA;k;n¼100;t¼ DA¼f ðP Þa;s;DA;k;n¼100;t¼ "
is then compared with Eq. (16). Fig. 1 shows the ratio of fractal dimension DP¼f ðAÞ=DA¼f ðP Þin relation with the experimental noise standard deviation for two ranges of sizes: 0.5 decade (Fig. 1a) and five decades for the area measurement (Fig. 1b) and six different theoretical fractal dimensions varying between 1 and 2. The simulated data (bold symbols) match very well with the probabilistic Eq. (16) (thin line).
Secondly the ratio of fractal dimension DP¼f ðAÞ=DA¼f ðP Þ is computed for all the configurations and compared with our probabilistic approach (Eq. (16)) that confirms our prior hypothesis to obtain an an-alytical formulation. A statistical analysis (analysis of variance) shows that the number of islands used to calculated the perimeter and the area is the most relevant factor which affects the difference between probabilistic and simulation results. Four configurations with n¼ 10–100 islands are considered in Fig. 2; it is shown that the coherence between the two approaches is quite good when the number of islands is higher than 50 notwithstanding a small bias for the lower values.
We then analysed the correlation DP¼f ðAÞa;s;DA;k;n¼100;t and DA¼f ðP Þa;s;DA;k;n¼100;t for a fractal dimension D¼ 1. For this purpose, DP¼f ðAÞa¼0:5;s;DA¼2;k;n¼100;t is plot versus DA¼f ðP Þa¼0:5;s;DA¼2;k;n¼100;t (Fig. 3a) for n¼ 100 simulations for
0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5
Noise standard deviation
D [ P=f (A )] / D [A=f (P ) ] D=1 D=1.2 D=1.4 D=1.6 D=1.8 D=2
Area Range = 0.5 decade
(a) 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 0 0.1 0.2 0.3 0.4 0.5
Noise standard deviation
D [ P=f (A )] / D [A=f (P )] D=1 D=1.2 D=1.4 D=1.6 D=1.8 D=2
Area Range = 5 decades
(b)
Fig. 1. Ratio of fractal dimension obtained by regression of P¼ f ðAÞ on fractal dimension obtained by regression A ¼ f ðP Þ versus the
experimental noise standard deviation of the perimeter measurement on the area measurement. These ratios are calculated by the probabilistic equation (16) (thin line) and by Monte-Carlo simulation (bold symbols are the means of 100 simulations). The number of decades for the area measurement is 0.5 (a) or 5 (b).
each different noise varying between 0.05 and 0.5. After linear regression, the slope between the two regressions is plotted versus the experimental noise, s, in Fig. 3b.
These surprising results can be shown:
ii(i) The slope decreases when the noise rises according to Eq. (16). For low noise, the correlation coeffi-cient r tends towards 1 and both Eqs. (3) and (8) give the same fractal dimension. For noises higher than 0.3 the slope becomes negative that cannot be accounted for Eq. (16). This equation assumes that the standard deviation for the noise is perfectly known and is not a random variable. However, this standard deviation obeys a determined statistical law. By 100,000 Monte-Carlo simulations for se0 ¼ 0:05 (positive correlation), se0¼ 0:3 (zero correlation) and se0 ¼ 0:5 (negative correlation), it was shown that the slope does not depend on the number of data considered for the regression, but that the scatter decreases as the number of data rises. In other words, the negative correlation does not disappear when the number of islands rises.
For a noise lower than 0.3, if DP¼f ðAÞrises, then DA¼f ðP Þrises. When the noise becomes higher than this critical value then if DP¼f ðAÞ rises, DA¼f ðP Þdecreases. As a consequence, for a noise higher than a critical value, when a physical process is positively correlated with the fractal dimension calculated by mean of the regression log(Perimeter) versus log(Area), then the same physical process is negatively correlated with the fractal dimension when the regression is performed on log(Area) versus log(Pe-rimeter). Some contradictions in literature are readily explained by this first result.
i(ii) The barycentre of each set of data corresponds to Eq. (16).
(iii) The distribution of the data around the DP¼f ðAÞis equal to 1. That is obvious since the Gauss–Markov hypotheses are verified for our simulation and therefore the fractal dimension calculated by the regres-sion log(Area) versus log(Perimeter) is unbiased and has 1 as average value.
To calculate the correlation between the two representations, a master curve has to be determined by successive simulations. With that aim in view we consider both regression and variance analysis. To begin, all configurations Pa;s;DA;k;n¼N ;tfor N ¼ f10; 20; 50; 100; 500g are simulated. Considering n higher than 50, it is possible to show that the slope only depends on the variable se=Da and on the fractal dimension. The master curves are determined by Monte-Carlo simulations on the entire domain@ (Fig. 4) supposing that the experimental work is performed with 50 islands whose surfaces vary within a two decades interval
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
D[ P=f(A) ] / D[ A=f(P) ] Simulated
D [ P=f(A) ] / D [ A=f(P ) ] Modelized n=10 n=20 n=50 n=100 ref
Fig. 2. Comparison between ratio of the fractal dimension calculated by P¼ f ðAÞ on A ¼ f ðP Þ. In ordinate, the ratio is obtained by
Eq. (16) and in abscissa by the Monte-Carlo simulation. All configurations of the set@ are considered and statistical analysis shows
that the parameter ‘‘number of islands’’ gives the biggest difference between analytical and Monte-Carlo models. The symbols are
related to the four sets of data (n¼ 10; . . . ; 100 islands) and the line represents a perfect correlation between the two models.
(DA¼ 2) and a se0¼ 0:2 standard deviation of the data versus the regression line. Two fractal dimensions are calculated with the two types of regression and the ratio DP¼f ðAÞ=DA¼f ðP Þ¼ 0:6 with se=Da¼ 0:1. From Fig. 4, it can be inferred that the fractal dimension is 1.2 if the experimental noise se0 is the same whatever the dimension of the island and without any noise in the area measurement.
3.2.5. Correlation between fractal dimension and physical process
In the above section, it was shown that a positive or a negative correlation might be found depending on the choice of the abscissa and ordinate. We now suppose that the fractal dimension is linearly well cor-related with a physical property. We shall determine if the island method is appropriate to calculate the fractal dimension according to the representation A¼ f ðP Þ or P ¼ f ðAÞ, the experimental noise and the correlation range. To this end, simulations are performed using the Monte-Carlo method.
Let the set of configurations be@a;s;DA;k;n;t, with amin2 f0:5; 0:6; . . . ; 1g;
a2 amin þ 0 na Da; aminþ 1 na Da; aminþ 2 na Da; . . . ; aminþ ka na Da; . . . ; aminþ na na Da ; D[ A=f(P) ] D [ P=f(A) ] 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 1 2 3 4 5 6 s=0.05 s=0.10 s=0.15 s=0.20 s=0.15 s=0.30 s=0.35 s=0.40 s=0.45 s=0.50 (a) Noise Measurement (s) Slope between D[ P=f(A) ] and D[ A=f(P) -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 Fractal dimension = 1
2 decades for area
Positive correlation
Negative correlation
(b)
Fractal dimension = 1
Fig. 3. (a) Monte-Carlo results DP¼f ðAÞa¼0:5;s;DA¼2;k;n¼100;t versus DA¼f ðP Þa¼0:5;s;DA¼2;k;n¼100;t, and (b) means of the ratio DP¼f ðAÞa¼0:5;s;DA¼2;k;n¼100;t on
DA¼f ðP Þa¼0:5;s;DA¼2;k;n¼100;t, for 100 simulations with noise, s, varying between 0.05 and 0.5 for a fractal dimension D¼ 1 and two decades for area measurement. The sign of the correlation depends on the experimental noise.
Da2 f0:1; 0:2; 0:5g; s 2 f0:01; 0:02; . . . ; 0:5g; DA2 f0:5; 0:1; . . . ; 5g; A2 0 nDA; 1 nDA; 2 nDA; . . . ; k nDA; . . . ; n nDA ;
nthe number of points used for the regression, A the area, a the slope of log P versus log A and t the index of the simulation.
Eq. (8) is simulated in the following form:
Stage 1: All pairs area–perimeter contained in the configuration set are simulated (Fig. 5).
Stage 2: The fractal dimensions DP¼f ðAÞ and DA¼f ðP Þ are calculated for all pairs of stage 1 by the least square method (Fig. 6).
Stage 3: The regression DP¼f ðAÞand DA¼f ðP Þare performed for values belonging to the set a. If the noise is low enough, then DP¼f ðAÞ DA¼f ðP Þ 2a. The real fractal dimension is plotted in Fig. 7 versus the estimated
Slope between D[ P=f(A) ] and D[ A=f(P) ] -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 D=1.0 D=1.2 D=1.4 D=1.6 D=1.8 D=2.0
Fig. 4. Master curve of the slope between DP¼f ðAÞand DA¼f ðP Þversus the ratio se0=Dafor fractal dimension varying between 1 and 2. All
Monte-Carlo configurations are considered.
Area (log) in pixel
Perimeter (log) in pixel 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 D=1 D=1.08 D=1.14 D=1.24 D=1.32 D=1.4 n=100 points
Fig. 5. First step of the Monte-Carlo algorithm: all pair log(Perimeter) versus log(Area) are simulated for fractal dimension varying between 1 and 1.4.
fractal dimension for the two regression modes. Fig. 7 requires a physical explanation. The abscissa rep-resents a fractal dimension varying between 1 and 1.4 and the ordinate the measure obtained by the re-gression of log(Perimeter) versus log(Area) on the left-hand size and log(Area) versus log(Perimeter) in the right-hand side of the graph. If the fractal dimension of the physical process rises linearly from 1 to 1.4 (amin¼ 0:5, Da¼ 0:2) and if the noise is low (se0¼ 0:01), then the two regressions give nearly the same slope and the representations DP¼f ðAÞand DA¼f ðP Þare equivalent. Considering now a higher noise (se0 ¼ 0:2) which corresponds to the experimental noise reported in literature, it can be shown that the correlation between the calculated and the theoretical fractal dimensions still holds for the first representation in spite of a rising scatter with the noise. On the contrary, if the regression is carried out on log(A) versus logðP Þ the slope is sometimes positive and sometimes negative. Consequently, the experimenter may find a downward trend in correlation of the fractal dimension with the physical process although the true correlation is positive. Worse than that, the classical statistic test might be significant.
Stage 4: Performance of the statistical calculation on the slope obtained in stage 3. D[ A=f(P) ] D [ P=f(A) ] 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 D=1.00 D=1.08 D=1.15 D=1.24 D=1.32 D=1.40 S=0.01 D[ A=f(P) ] D [ P=f(A) ] 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 S=0.05 D=1.00 D=1.08 D=1.15 D=1.24 D=1.32 D=1.40 D[ A=f(P) ] D [ P=f(A) ] 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2 4 6 8 10 12 S=0.3 D=1.00 D=1.08 D=1.15 D=1.24 D=1.32 D=1.40 (a) (b) (c)
Fig. 6. Second step of the Monte-Carlo algorithm: The slope of DP¼f ðAÞ versus DA¼f ðP Þ is simulated for three values of the residual
The histograms in Fig. 8 represent the distribution of the slope for the two representations. It is shown that:
ii(i) The scatter in the slope rises with se0.
i(ii) The histograms are symmetrical for the DP¼f ðAÞand seem Gaussian.
(iii) The more the dispersion rises, and the more the distribution of DA¼f ðP Þloses its Gaussian aspect. (iv) The values are not centred on 1 for DA¼f ðP Þ and this shift is higher when the noise residual standard
deviation rises.
We now calculate the rank correlation instead of the statistic moments since they are not defined for calculating DA¼f ðP Þ. We only calculate the median and the 5% and 95% quantiles since they are always defined (Fig. 9).
Stage 5: The correlation diagram is plotted to answer the question: is the correlation DA¼f ðP Þsignificant? As the indicator of correlation we have chosen is the median, we shall use two parametric statistical tests (rank and sign test). The critical probability we plot in ordinate is the probability that we wrongly state by the median equals zero, i.e. the correlation between the fractal dimension and the physical process does not exist. As usual, the threshold 0.05 is considered. The graphs have been plotted (Fig. 10) for the two types of representation. It can easily be shown that the correlation becomes insignificant (95% confidence interval) if the noise residual standard deviation varies between 0.16 and 0.18. If se0 <0:16, the correlation is positive and if se0>0:18, it is negative.
In other words, if a physical process is perfectly positively correlated with the fractal dimension which varies between 1 and 1.4, then the experimenter who uses the representation A¼ f ðP Þ, may find according
Theoretical fractal dimension
D[ P=f(A) ] 0.9 1 1.1 1.2 1.3 1.4 1.5 1 1.1 1.2 1.3 1.4 S=0.01
Theoretical fractal dimension
D[ A=f(P) ] 0.9 1 1.1 1.2 1.3 1.4 1.5 1 1.1 1.2 1.3 1.4 S=0.01
Theoretical fractal dimension
D[ P=f(A) ] 0.5 1 1.5 2 2.5 3 3.5 4 0.95 1.05 1.15 1.25 1.35 1.45 S=0.20
Theoretical fractal dimension
D[ A=f(P) ] 0.5 1 1.5 2 2.5 3 3.5 4 1 1.1 1.2 1.3 1.4 S=0.20
Fig. 7. Correlation between simulated and theoretical fractal dimension by regressing perimeter versus the area (left) or area versus
perimeter (on the right). Three residual standard deviation (se0¼ 0:01, 0.05, 0.2) are considered. One line corresponds to a Monte-Carlo
simulation. 1094
to the experimental error (se0) if the correlation is positive, negative, or does not exist. If the representation P¼ f ðAÞ is preferred, he will always obtain a positive correlation. Nevertheless, this numerical application with mean values chosen among the results coming from the literature depends on amin, Da, DA. This simulation has to be extended to all the configurations of the simulation set, and after calculation it has been shown [42] that the median can be modelled by the following equations:
MedA¼f ðP Þ¼ 1 726½1 0:910:004amin 0:420:006Da se DA 20:02 ð17Þ MedP¼f ðAÞ¼ 1 ð18Þ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 0 0.1 0.2 0.3
True slope / regression slope
IC [A= f(P)] IC [A= f(P)] IC [P= f(A)] IC [P= f(A)] Median[A= f(P)] Median[P= f(A)]
Fig. 9. No-parametrical statistic analysis of the histograms in Fig. 8. IC represents the 95% confidence interval.
Frequency 0 1000 2000 3000 4000 5000 6000 7000 -3 -2 -1 0 1 2 3 0 2000 4000 6000 8000 -3 -2 -1 0 1 2 3 2 3 2 3 0 400 800 1200 1600 2000 2400 2800 -3 -2 -1 0 1 2 3 0 400 800 1200 1600 2000 2400 2800 -3 -2 -1 0 1 A=f(P) 0 150 300 450 -3 -2 -1 0 1 2 3 P=f(A) 0 100 200 300 400 500 600 700 -3 -2 -1 0 1 S=0.01 S=0.01 S=0.05 S=0.05 S=0.2 S=0.2
Fig. 8. Histograms of the slope between simulated versus calculated fractal dimensions obtained for 10,000. Monte-Carlo simulations
and a residual standard deviation on the area or perimeter measurement se0¼ 0:01, 0.05, 0.2. Right are the results for the P ¼ f ðAÞ
Eqs. (17) and (18) represent the influence of the abscissa and ordinate chosen for the SIM representation on the correlation under the assumption that a uniform noise affects only the measurement of the perimeter. To summarise the physical aspect of this development we consider a relation between the fractal di-mension and the fracture toughness such that KIC¼ 50D. The fractal dimension varies from 1 to 1.4 (amin¼ 0:5, Da¼ 0:2) the standard deviation in the perimeter determination is 0.2 lm (se0 ¼ 0:2) and the area is measured on 1 decade (DA¼ 0:2). We find MedA¼f ðP Þ¼ 0:32, and a 50% probability that the correlation is negative by regressing area versus perimeter.
4. Application and discussion
It was shown that the relation between the theoretical and the experimental fractal dimension DP¼f ðAÞis unbiased if there is no noise when recording the area. Only the amplitude of the experimental noise, a low range for the area variation, a low variation between D and the physical process or a reduced number of recorded islands prevent from calculating the precise fractal dimension. If a relation is found, then this relation has a physical signification.
To test whether the experimental noise has an effect on the result, we now simulate the fractal dimension by a Monte-Carlo algorithm and the bias obtained by regression A¼ f ðP Þ and P ¼ f ðAÞ by considering sP and sAthe standard deviation for the perimeter and the area respectively. We also propose a new method to perform the regression by minimising the orthogonal distance between experimental data and the slope (principal component analysis) instead of vertical or horizontal distance. This method is rather difficult to compute since the eigenvalues have to be calculated. However it gives a fractal dimension which is inde-pendent of the type of representation A¼ f ðP Þ or P ¼ f ðAÞ.
Three limited cases are considered in Fig. 11:
ii(i) Noise is introduced only on the perimeter measurement. i(ii) Noise is introduced on both perimeter and area measurement. (iii) Noise is introduced only on the area measurement.
From Fig. 11, it can be inferred:
ii(i) DP¼f ðAÞis unbiased since the area is recorded without any noise. i(ii) DA¼f ðP Þis unbiased since the perimeter is recorded without any noise.
1E-65 1E-60 1E-55 1E-50 1E-45 1E-40 1E-35 1E-30 1E-25 1E-20 1E-15 1E-10 1E-05 1 0 0.1 0.2 0.3 0.4 0.5
Standard error of noise measurements
Critical probability Null correlation Positive correlation Negative correlation
Fig. 10. Critical probability versus the experimental noise for the two type of regression (a) A¼ f ðP Þ and (b) P ¼ f ðAÞ. The data are
related to the configuration: a2 f0:5g; Da2 f0:2g; DA2 f1g, and n ¼ 100.
(iii) The orthogonal regression is unbiased if the perimeter and area scatter is of the same order of magni-tude.
(iv) The DA¼f ðP Þbias rises with the fractal dimension, accordingly to Eq. (12).
i(v) The DA¼f ðP Þbias is higher than the DP¼f ðAÞbias if the uncertainty on the perimeter and area determina-tions is of the same order of magnitude.
(vi) The fractal dimension calculated by the orthogonal method lies between the two other results. We then compute the fractal dimension by considering sP and sA the standard deviations for the pe-rimeter and the area. We must first quantify the error made by the representation DP¼f ðAÞwith a noise sA which violates the Gauss–Markov hypothesis. After calculations, it can be shown that:
BiasP¼f ðAÞ¼ 3:30:01D sA DA
1:50:0015
ð19Þ Theoretical Fractal Dimension
Calculated fractal dimension 1 1.2 1.4 1.6 1.8 2 2.2 2.4 1 1.2 1.4 1.6 1.8 2 2 2 P=f(A) A=f(P) Orthogonal (a) s A=0 sP=0.1
Theoretical Fractal Dimension
Calculated fractal dimension 1 1.2 1.4 1.6 1.8 2 2.2 2.4 1 1.2 1.4 1.6 1.8 P=f(A) A=f(P) Orthogonal sA=0.1 sP=0.1 (b)
Theoretical Fractal Dimension
Calculated fractal dimension 1 1.2 1.4 1.6 1.8 2 2.2 2.4 1 1.2 1.4 1.6 1.8 P=f(A) A=f(P) Orthogonal sA=0.1 sP=0 (c)
Fig. 11. Fractal analysis with the three regressions: A¼ f ðP Þ, P ¼ f ðAÞ and orthogonal. sAand sP represent respectively the noise on
area and perimeter measurements. The symbols represent the average of 100 Monte-Carlo simulations with DA2 f0:5; 0:1; . . . 5g and a
number of data 50, 250 and 450. The line represents the perfect relation between the theoretical and the calculated fractal dimensions and is given as guidelines for the eyes.
This relation model the error made on the fractal dimension with a0.0395%uncertainty. It can be shown that the bias is independent of both the number of islands considered and the standard deviation.
The same process gives a negative bias considering the DA¼f ðP Þ representation: BiasA¼f ðP Þ¼ 51:120:12 1 D sP DP 2:040:0013 ð20Þ From these equations, it is shown that an increasing noise on the area measurement induces a decreasing calculated fractal dimension. In the same way, the fractal dimension becomes higher if the number of decade rises.
By non-linear regression, the orthogonal representation induces the following bias:
BiasOrtho¼ 1:290:11D sP DP s A DA 19:60:26 sP DP 2:150:01 þ 5:930:17 sP DP 1:910:02 þ 6:080;:8 sP DP s A DA ð21Þ
These three preceding equations allow us to calculate the error made on the fractal dimension by using the slit island method if the standard deviations on the perimeter and area measurement are known.
To test the efficiency of (19)–(21) we draw in Fig. 12 the modelled errors related to our experimental values. It is obvious that:
ii(i) Our models well represent statistical errors in the SIM.
i(ii) It avoids using the A¼ f ðP Þ representation since the bias is too important, moreover it is always neg-ative and gives a fractal dimension higher than the true one.
(iii) Using the P ¼ f ðAÞ relation always leads to a positive bias. That means that the calculated fractal di-mension is always less than the true one. That might explain why Ray et al. found a negative fractal dimension using that representation [6].
(iv) Choice of orthogonal or P ¼ f ðAÞ regression depends on the formula which minimise the statistical bias.
Moreover Eqs. (19)–(21) give possible explanations for negative or positive relationship reported be-tween fractal the dimension and mechanical properties.
As an example of the use of these relations we consider the two following cases supposing that no re-lations exists between the fractal dimension and MP.
A negative correlation can be found by regressing perimeter versus the area if:
i(i) The standard deviation in the area measurement is more important when MP increases. (ii) The number of decades rises when MP becomes higher.
A positive correlation can be found if:
i(i) The perimeter standard deviation is higher when MP increases. (ii) The number of decade decreases with MP.
This conclusion is not only a theoretical case with no probability to occur. Literature results show that the island morphology (area, shape . . .) evolves with mechanical properties and then an erroneous corre-lation can be found with sA=DA (in Eq. (19)) or sP=DP (in Eq. (20)).
We search in the literature results containing sufficient information to carry out statistical simulations about the variation of the fractal dimension of fractured surface in relation with mechanical properties. It was found that:
Calculated bias Modeled bias -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 RESIDUS 0 500 1000 1500 2000 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 P = f(A) Residuals Calculated bias Modeled bias -5 -4 -3 -2 -1 0 1 -5 -4 -3 -2 -1 0 Residus N o of obs 0 1000 2000 3000 4000 5000 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 A = f(P) Residuals Calculated bias Modeled bias -1.8 -1.4 -1 -0.6 -0.2 0.2 0.6 1 -2 -1.4 -0.8 -0.2 0.4 1 RESIDUS 0 20 40 60 80 -1.5 -1 -0.5 0 0.5 1 Orthogonal Residuals
Fig. 12. Comparison of modelled and simulated bias obtained for the three representations P¼ f ðAÞ (Eq. (19)), A ¼ f ðP Þ (Eq. (20))
ii(i) The scatter in area and perimeter measurements are always the same, respectively 0.15 (with a standard deviation of r¼ 0:05) and of 0.11 (r ¼ 0:07) which is statistically equivalent (Student test for results reported in Table 1).
i(ii) Residuals obtained for area perimeter representation are Gaussian. (iii) Many errors are made on the calculation of the fractal dimension.
As a consequence, it is difficult to state if positive or negative correlation can be deduced from the lit-erature results between the fractal dimension and mechanical properties (or other physical processes) since the experimental noise introduces artefacts both in calculation of the fractal dimension and in the corre-lation.
Many researchers have tried to correlate the fractal dimension with mechanical properties such as impact energy and fracture toughness from the observation of surface of the fracture. Some researchers report the validity of the fractal analysis but other researchers report the converse results. The main reason for the discrepancy is that the actual fracture mechanisms and processes are very complex and not so ideal for the fractal analysis to be easily applied. In some cases, different mechanisms control the fracture and the linear relation is not obtained on log–log plots of data obtained from mechanical tests with a wide range of test temperatures, heat treatment, etc. We have proposed a statistical reason for the discrepancy. However, we have supposed that data perfectly obeys the Gauss–Markov hypotheses that will be false if the linear re-lation is rejected. What will our conclusion be if data violate the linear rere-lation? Firstly, if the linear rere-lation is rejected then the fractal dimension cannot be estimated formally. However it is sure that a low deviation from the linear relation can be observed without systematically rejecting the fractal concept. In this case, what will be the evaluation of the fractal dimension? Does a statistical artefact then appear?
To appreciate this possible artefact, one has to introduce a non-pure linear relation characterizing the usual non-linear representation met in the bibliography.
P ¼ 0:75hA þ eðlog A aðh; eÞÞ ð22Þ
where aðh; eÞ is calculated such that the fractal dimension 2 0:75hA will be calculated without bias by the least square regression using the P ¼ f ðAÞ representation. P ¼ 0:75hA þ eðlog A aðh; eÞÞ is an amplified factor of the non-linearity and h is a variable that will change the fractal dimension.
Fig. 13 represents the log–log graph for different e values and h¼ 1. We have then calculated the value of the fractal dimension by the representation A¼ f ðP Þ. We obtained the following results:
• For any fixed e, if h increases (respectively decreases), then the fractal dimension DA¼f ðP Þincreases (re-spectively decreases): If h is a physical parameter that modifies the fractal dimension, a constant change of linearity does not affect this tendency and then no false correlation is introduced (see Fig. 14). • The non-linearity does not really change the results we present in this article about the influence of the
statistical noise on the area and perimeter measurement on the fractal dimension.
• The fractal dimension depends on the e values. Fig. 15 represents the effect of the non-linearity on DA¼f ðP Þ. If non-linearity increases, the calculated fractal dimension decreases: it is an important fact be-cause if fracture mechanisms or processes involve a monotonous change with a physical parameter (for example test temperature, heat treatment) then a correlation can be found between the mechanical prop-erties and the fractal dimension even if no correlation exists. However, all simulations we have carried out seem to converge to the opinion that non-linearity will not affected the determination of the fractal dimension by a factor higher than 0.02 (see Fig. 15).
However, to us whatever the representation used, the highest artefact could result from the change of sA, DA, sP or DP with the physical process according to Eqs. (19)–(21). Let us illustrate the example met in the bibliography [44] to which we can apply our results. Using the DP¼f ðAÞrelationship, Su et al. [44] find 1100
log(Area) log(Perimeter) 0 0.5 1 1.5 2 2.5 3 3.5 4 0.5 1.5 2.5 3.5 4.5 5.5 e 0 0.2 0.4 0.6 0.9
Fig. 13. Log–log graph perimeter–area relation for different e values such that P¼ 0:75A þ eðlog A að1; eÞÞ.
D[A=f(P)] 1.5 1.55 1.6 1.65 1.7 1 1.02 1.04 1.06 1.08 1.1 e 0 0.2 0.4 0.6
Fig. 14. Evolution of DA¼f ðP Þwith a physical parameter h given by Eq. (22) with different e values.
D[A=f(P)] 1.486 1.488 1.49 1.492 1.494 1.496 1.498 1.5 0 0.2 0.4 0.6 0.8 1
that the fractal dimension of pearlite in HSLA steel increases from 1.47 to 1.57 with different normalizing temperatures (900–1200 C). The higher normalizing temperature, the higher the pearlite area (increases of the radius size from 20 to 50 lm quasi-linearly). This means that DA will double and according to Eq. (19) with supposing sA constant introduced a relation of bias of to 0.03–0.09. We have simulated under the condition given by Su et al. supposing that the fractal dimension does not depend on the normalizing time and equals 1.7, the fractal dimension calculation versus the normalizing time and reported their results. As can be observed in Fig. 16, even if the fractal dimension does not change with the normalizing time, an increasing tendency of fractal dimension with the normalizing time is found near the relation obtained by Su et al. and only due to the statistical bias introduced by the SIM. That does not mean that the relation claimed by the authors does not exist but simply means that the bias will emphases this relation.
5. Conclusion
Statistical analysis and simulation have been used to study the fractal dimension calculated by the SIM. The analysis shows that the results depend on the choice of ordinates and abscissa chosen for the area and the perimeter. If there is no noise on the area, then the representation P ¼ f ðAÞ gives a correct value of D. If an experimental noise exists on the perimeter, the regression A¼ f ðP Þ gives erroneous value for D, espe-cially if the perimeter range decreases and the noise becomes more important. For this case, the calculated fractal dimension is higher than the true one. If the measurement is of the same order of magnitude as far as area and perimeter are concerned, then the new orthogonal method we proposed is more suitable to cal-culate D.
Finally, it was shown that the SIM might produce artefacts in the correlation between fractal dimension and a physical process, and that some results in literature should be considered with much caution.
The main defect of the SIM is its lack of statistical robustness to study the fractal dimension related to a physical process. Some other artefacts can occur to modify the information measurement during image acquisition (focusing, threshold, magnification, brightness . . .) and give erroneous results. Moreover, the authors who used the classical statistical test for linear regression in log–log co-ordinates give a fractal dimension D¼ a b. The uncertainty b is generally low, but this method supposes that the Gauss–Markov
Normalising temperature (˚C) Fractal dimension 1.45 1.5 1.55 1.6 900 950 1000 1050 1100 1150 1200 Simulation Su et al.
Fig. 16. Effect of the bias introduced by the Slit Island Method when fractal dimension stays unchanged and equals 1.7 (thin line) and experimental data given by Su et al.
hypotheses are verified, and we have shown that current statistics can give false values. Consequently, the attempt to correlate the fractal dimension with a physical process first requires a good unbiased estimation of D.
Acknowledgement
We wish to thank Veeronique Hague for her assistance in English. Appendix A
The T truncated Gaussian law is defined by: uðtÞ ¼ fðtÞ
2FðtÞ 1 for T 6 t 6 T ðA:1Þ
and uðtÞ ¼ 0 elsewhere with fðtÞ ¼ 1ffiffiffiffiffiffi 2p p exp t 2 2 and FðX Þ ¼ Z X 1 fðtÞ dt The moments (null for k odd) are estimated by:
lk ¼ Z T T tuðtÞ ¼ ðk 1Þlk2 2fðT Þ 2FðT Þ 1T k1 ðA:2Þ
As l0¼ 1, Eq. (A.2) can be written thanks to an expansion as: lk ¼ 1:3 . . . ðk 1Þ 1 2fðT Þ 2FðT Þ 1 T þT 3 3 þ T5 3:5þ þ Tk1 3:5 . . .ðk 1Þ However 2fðT Þ 2FðT Þ 1¼ T þT 3 3 þ T5 3:5þ and then: lk ¼ 2fðT Þ 2FðT Þ 1 Tkþ1 ðk þ 1Þ 1 þ T 2 kþ 3þ T4 ðk þ 3Þðk þ 5Þþ
To converge we must have: lim k!1 lkþ2 lk <1 and lim k!1 lkþ2 lk < T2 ðA:3Þ Then T < 1.
Eq. (10) converges if:
lim k!1 ra a a kþ2 lkþ2 ra a a k lk ¼ ra a a 2 lim k!1 lkþ2 lk <1 ðA:4Þ
Then from Eq. (A.3), Eq. (A.4) will converge if the reduced centred Gaussian
x¼ 2 DA¼f ðP Þ aa
ra
is truncated by the T variable under condition given by Eq. (A.3) and then including Eq. (A.4), we get the convergence ifðra=aaÞ
2 T2<1.
References
[1] Mandelbrot BB. Les objets fractals: forme, hasard et dimension. Paris: Flammarion; 1975. [2] Mandelbrot BB. The fractal geometry of nature. New York: W.H. Freeman; 1983.
[3] Bouchaud E, Lapasset G, Planees J, Naveos S. Statistic of branched fracture surfaces. Phys Rev B 1993;48(5):2917–28. [4] Bouchaud JP, Bouchaud E, Lapasset G, Planes J. Models of fractal cracks. Phys Rev Lett 1993;71(14):2240–3. [5] Saouma VE, Barton C. Fractals, fractures, and size effects in concrete. J Engng Mech 1994;120(4):835–54.
[6] Ray KK, Mandal G. Study of correlation between fractal dimension and impact energy in a high strength low alloy steel. Acta Metall Mater 1992;40(3):463–9.
[7] Weisheng L, Bingsen C. Discussion on ‘‘the fractal effect of irregularity of crack branching on the fracture toughness of brittle materials’’ by Xie Heping. Int J Fract R 1994:R65–70.
[8] Wang ZG, Chen DL, Jiang XX, Ai SH, Shih CH. Relationship between fractal dimension and fatigue threshold value in dual-phase steels. Scripta Met 1988;22:827–32.
[9] Charkaluk E, Bigerelle M, Iost A. Fractal and rupture. Engng Fract Mech 1998;61(1):119–39.
[10] Tanaka M. Fracture toughness and crack morphology in indentation fracture of brittle materials. J Mater Sci 1996;31:749–55. [11] Mecholsky JJ, Passoja DE, Feinberg-Ringel KS. Quantitative analysis of brittle fracture surfaces using fractal geometry. J Am
Ceram Soc 1989;72(1):60–5.
[12] Mecholsky JJ, Mackin TJ. Fractal analysis of fracture in Ocala chert. J Mater Sci Lett 1988;7:1145–7.
[13] Lange DA, Jennings HM, Shah SP. Analysis of surface roughness using confocal microscopy. J Mater Sci 1993;28:3879–84. [14] Mandelbrot BB, Passoja DE, Paullay AJ. Fractal character of fracture surfaces of metals. Nature 1984;308:721–2.
[15] Su H, Zhang Y, Yan Z. Fractal analysis of microstructures and properties in ferrite–martensite steels. Scripta Metall Mater 1991;25:651–4.
[16] Lin G, Lai JKL. Fractal characterization of fracture surfaces in a resin-based composite. J Mater Sci Lett 1993;12:470–2. [17] Saouma VE, Barton C. Fractal characterization of fractured surfaces in concrete. J Engng Fract Mech 1990;35:47–53. [18] Pande CS, Richards LE, Louat N, Dempsey BD, Schwoeble AJ. Fractal characterization of fractured surfaces. Acta Met
1987;35(7):1633–7.
[19] Richards LE, Dempsey BD. Fractal characterization of fractured surfaces in Ti–4.5Al–5.0Mo–1.5Cr (CORONA 5). Scripta Met 1988;22:687–9.
[20] Brandt AM, Prokopski G. On the fractal dimension of fracture surfaces of concrete elements. J Mater Sci 1993;28:4762–6. [21] Bouchaud E, Lapasset G, Planes J. Fractal dimension of fractured surfaces: a universal value? Europhys Lett 1990;13(1):73–9. [22] Underwood EE, Banerji K. Fractals in fractography. Mater Sci Engng A 1986;80:1–14.
[23] Mandelbrot BB. How long is the coast of Britain? Statistical self-similarity and fractal dimension. Science 1967;155:636.
[24] Minkowski H. Uber die Begriffe L€aange, Oberfl€aache und Volumen. Deut Math 1901;Jahr9:115–21.
[25] Tricot C. Courbes et dimension fractale. Paris: Springer-Verlag; 1993.
[26] Dubuc B, Quiniou JF, Roques-Carnes C, Tricot C, Zucker SW. Evaluating the fractal dimension of profiles. Phys Rev A 1989;39(3):1500–12.
[27] Ganti S, Busham B. Generalized fractal analysis and its applications to engineering surfaces. Wear 1995;180:17–34. [28] Lovejoy S. Area–perimeter relation for rain ans cloud areas. Science 1982;216:185–7.
[29] Elias H, Schwartz D. Surface areas of the cerebral cortex of mammals. Science 1969;166:111–3. [30] Weibel ER. Morphometry of the human lung. New York: Academic Press; 1963.
[31] Hack JT. Studies of longitudinal streams in Virginia and Maryland. US Geological Survey Professional Papers, vol. 294B, 1957. [32] Huang ZH, Tian JF, Wang ZG. A study of the slit island analysis as a method for measuring fractal dimension of fractured
surface. Scripta Metall Mater 1990;24:967–72.
[33] Mc-Anulty P, Meisel LV, Cote PJ. Hyperbolic distributions and fractal character of fracture surfaces. Phys Rev A 1992;46(6):3523–6.
[34] Imre A. Problems of measuring the fractal dimension by the slit-island method. Scripta Metall Mater 1992;27:1713–6. 1104
[35] Imre A. Comment on ‘‘perimeter-maximum-diameter’’ method for measuring the fractal dimension of a fracture surface. Phys Rev B 1995;51(22):16470.
[36] Mu ZQ, Lung CW, Kang Y, Long Q. Perimeter-maximum method for measuring the fractal dimension of a fractured surface. Phys Rev B 1993;48(10):7679–81.
[37] Chen Z, Mecholsky JJ, Joseph T, Beatty CL. The fractal geometry of Si3N4 wear and fracture surfaces. J Mater Sci 1997;32:6317– 23.
[38] Lung CW, Mu ZQ. Fractal dimension measured with perimeter–area relation and toughness of materials. Phys Rev B 1988;38(16):11781–4.
[39] Lung CW, Zhang SZ. Fractal dimension of the fractured surface of materials. Physica D 1989;38:242–5.
[40] Meisel LV. Perimeter–area analysis, the slit-island method and the fractal characterization of metallic fracture surfaces. J Phys D: Appl Phys 1991;24:942–52.
[41] Bigerelle M, Iost A. Analysis of the Koch island perimeter by the Richardson method. Monte-Carlo application to grain boundaries evolution during heating, in press.
[42] Bigerelle M. PhD thesis, Caracterisation geeomeetrique des surfaces et interfaces, applications en meetallurgie. ENSAM Lille, 1999.
[43] Schneider JM, Bigerelle M, Iost A. Statistical analysis of the Vickers hardness. Mat Sci Eng A 1999;262:256–63.
[44] Su H, Yan Z, Stanley JT. Fractal analysis of microstructure and properties of HSLA steels. J Mater Sci Lett 1995;14:1436–9. [45] Pande CS, Richards LR, Smith S. Fractal characteristics of fractured surfaces. J Mater Sci Lett 1987;6:295–7.
